Erratum to: An $L_{p}$-theory of Stochastic PDEs of Divergence Form on Lipschitz Domains

Abstract

The purpose of this erratum is to correct the assumptions in Theorem 2.10 of [2] (Kim in J Theor Probab 22:220–238, 2009).

Erratum to: J Theor Probab (2009) 22:220–238 DOI 10.1007/s10959-008-0170-x

Theorem 2.10 of [2] should be corrected as follows. The point here is that condition (0.1) below is stronger than the one assumed in [2].

Theorem 0.1

Let $p\in [2,\infty )$ and Assumptions 2.1, 2.3, 2.4, 2.7 and 2.8 be satisfied. There exists $\kappa =\kappa (\delta _0,p,d,K)\in (0,1)$ such that if

$$\begin{aligned} d-2+p-\kappa <\theta <d-2+p+\kappa \end{aligned}$$

(0.1)

then for any $f\in \psi ^{-1}{\mathbb {H}}^{-1}_{p,\theta }({\mathcal {O}},\tau ), f^i\in {\mathbb {L}}_{p,\theta }({\mathcal {O}},\tau ), g\in {\mathbb {L}}_{p,\theta }({\mathcal {O}},\tau )$ and $u_0\in U^{1}_{p,\theta }({\mathcal {O}})$, Eq. (1.1) with initial data $u_0$ has a unique solution $u \in {\mathfrak {H}}^{1}_{p,\theta }({\mathcal {O}},\tau )$, and for this solution

$$\begin{aligned} \Vert u\Vert _{{\mathfrak {H}}^{1}_{p,\theta }({\mathcal {O}},\tau )}\le N \left( \Vert \psi f\Vert _{{\mathbb {H}}^{-1}_{p,\theta }({\mathcal {O}},\tau )}+\Vert f^i\Vert _{{\mathbb {L}}_{p,\theta }({\mathcal {O}},\tau )} +\Vert g\Vert _{{\mathbb {L}}_{p,\theta }({\mathcal {O}},\tau )} +\Vert u_0\Vert _{U^{1}_{p,\theta }({\mathcal {O}})}\right) , \end{aligned}$$

(0.2)

where $N=N(d,p,\delta _0,K,T,{\mathcal {O}})$.

In Theorem 2.10 of [2], in place of (0.1), the weaker condition

$$\begin{aligned} d-\kappa <\theta <d-2+p+\kappa \end{aligned}$$

(0.3)

is assumed.

The error of the proof of [2, Theorem 2.10] occurred because it relied on a result proved in [3, Theorem 2.1], which is related to non-divergence type SPDE. The result of [3, Theorem 2.1] is proved for the range of $\theta$ satisfying (0.3), but it turns out that [3, Theorem 2.1] is false unless much stronger restriction on $\theta$ is assumed (see [1] for details).

Theorem 2.1 of [3] is corrected in [1, Theorem 2.12] for $\theta$ satisfying (0.1). Thus the proof of Theorem 2.10 of [2] goes throughout without any change if condition (0.1) is assumed.